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Operator theoretic characterizations of $ [IN]$-groups and inner amenability


Authors: Anthony To Ming Lau and Alan L. T. Paterson
Journal: Proc. Amer. Math. Soc. 102 (1988), 893-897
MSC: Primary 43A15; Secondary 22D25, 43A07, 47B38
DOI: https://doi.org/10.1090/S0002-9939-1988-0934862-0
MathSciNet review: 934862
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Abstract: Let $ G$ be a locally compact group and $ p \in [1,\infty ]$. Let $ {\pi _p}$ be the isometric representation of $ G$ on $ {L_p}(G)$ given by $ {\pi _p}(x)f(t) = f({x^{ - 1}}tx)\Delta {(x)^{1/p}}$. Let $ {\mathcal{A}'_p}$ be the commutant of $ {\mathcal{A}_p}$ in $ B({L_p}(G))$. In this paper we determine those $ G$ for which: (*) $ {\mathcal{A}'_p}$ contains a nonzero compact operator. We prove, among other things, that if $ p \in [1,\infty )$, then (*) holds if and only if $ G \in [IN]$, and that if $ p = \infty $, then (*) holds if and only if $ G$ is inner amenable.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0934862-0
Keywords: Locally compact groups, $ [IN]$-groups, fixed-points, operator algebras, compact operators, inner invariant means, amenability
Article copyright: © Copyright 1988 American Mathematical Society

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